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The idea of crystalline cohomology in characteristic 0 is to find a direct definition of a cohomology theory as the cohomology of constant sheaves on a suitable site. The site Inf X is a category whose objects can be thought of as some sort of generalization of the conventional open sets of X. Grothendieck showed that for smooth schemes X over C , the cohomology of the sheaf O X on Inf X is the same as the usual smooth or algebraic de Rham cohomology.
In characteristic p the most obvious analogue of the crystalline site defined above in characteristic 0 does not work. Grothendieck solved this problem by defining objects of the crystalline site of X to be roughly infinitesimal thickenings of Zariski open subsets of X , together with a divided power structure giving the needed divided powers.
More generally one can work over a base scheme S which has a fixed sheaf of ideals I with a divided power structure. A key point of the theory is that the crystalline cohomology of a smooth scheme X over k can often be calculated in terms of the algebraic de Rham cohomology of a proper and smooth lifting of X to a scheme Z over W.
There is a canonical isomorphism. Conversely the de Rham cohomology of X can be recovered as the reduction mod p of its crystalline cohomology after taking higher Tor s into account. This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology. The term crystal attached to the theory, explained in Grothendieck's letter to Tate , was a metaphor inspired by certain properties of algebraic differential equations.
These had played a role in p -adic cohomology theories precursors of the crystalline theory, introduced in various forms by Dwork , Monsky , Washnitzer, Lubkin and Katz particularly in Dwork's work. Such differential equations can be formulated easily enough by means of the algebraic Koszul connections , but in the p -adic theory the analogue of analytic continuation is more mysterious since p -adic discs tend to be disjoint rather than overlap.
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By decree, a crystal would have the 'rigidity' and the 'propagation' notable in the case of the analytic continuation of complex analytic functions. From Wikipedia, the free encyclopedia.
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